Can neural network help me with detecting center coordinates of particles in an image?
This is a very hard problem, you have many overlapping points with objects which aren't completely round. I'm not very knowledgeable on CV but I suspect you will find it very challenging.

I would probably say a handcrafted detection algorithm would probably be easier, something like an edge detector which fit circles to arcs and labeled the points. But it's still going to be nontrivial and maybe impossible to get it working with high accuracy.